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Preprint Title The influence of the magnetic nanoparticles coating from colloidalsystem on the magnetic relaxation time

Authors Mihaela Osaci and Matteo Cacciola

Publication Date 06 Dez 2019

Article Type Full Research Paper

ORCID® iDs Mihaela Osaci - https://orcid.org/0000-0003-4062-7556

1

The influence of the magnetic nanoparticles coating from colloidal system on the magnetic

relaxation time

Mihaela Osaci*

“Politehnica” University of Timisoara,

Department of Electrical Engineering and Industrial Informatics,

2 Victoriei Square, 300006 Timisoara, Timis County, Romania

*corresponding author, email: mihaela.osaci@fih.upt.ro

Matteo Cacciola

Cooperativa TEC, Via Nazionale, n. 439, I-89134 Pellaro di Reggio Calabria

Abstract

Currently, the colloidal systems of monodomain superparamagnetic nanoparticles are used in

biomedical applications, such as hyperthermia treatment for cancer. In this type of colloid, called

nanofluid, there is a tendency of nanoparticle agglomeration. It has been shown experimentally that the

nanoparticle coating plays an important role in the nanoparticle dispersion stability and

biocompatibility, which role was theoretically understudied so far. Also, the implications of

nanoparticle coating on the magnetic properties of the nanoparticles have been little studied. This

paper presents the theoretical study, by numerical simulation, on how the nanoparticle coating affects

the tendency of agglomeration of the nanoparticles and the Néel relaxation time – or the effective

magnetic relaxation time of the system. For simulating the self-organization of colloidal nanoparticles,

we apply a Langevin dynamics stochastic method based on an effective Verlet-type algorithm, and the

Néel magnetic relaxation time is assessed through the Coffey method in oblique magnetic field,

adapted to the local magnetic field on a nanoparticle.

2

Keywords: magnetic nanoparticles coating, colloidal system, magnetic relaxation time,

simulation, Langevin dynamics stochastic method, effective Verlet-type algorithm

Introduction

One of the most important biomedical applications of the colloidal magnetic nanoparticle

systems is the magnetic hyperthermia as an alternative cancer treatment. The magnetic nanoparticles,

after reaching the tumour, are locally subjected to an alternative magnetic field and they generate heat

that kills the cancer cells [1], in general through two phenomena: Néel relaxation (internal

phenomenon, i.e. rotation of the particle magnetic moment inside the particle) and Brown relaxation

(external phenomenon, i.e. rotation of the nanoparticle along with the magnetic moment). Both internal

and external sources of “friction” lead to a delay phase in the orientation of the particle magnetic

moments in the direction of the applied magnetic field, thus generating heat. This heat increases the

temperature of the tumour cell and, ultimately, kills it [1- 4].

The magnetic nanoparticles, currently studied intensely for applications in magnetic

hyperthermia, are iron oxide nanoparticles, in particular magnetite (Fe3O4) and maghemite (γ-Fe2O3),

due to the possibilities of obtaining them in small dimensions for their reduced toxicity and possibilities

of easy functionalization of their surfaces. Some of the methods currently used to synthesize these

nanoparticles are: the modified sol-gel method used to prepare nanoparticles of nanometric size under

supercritical conditions of ethyl alcohol and alkaline co-precipitation, with an additional step of a

hydrothermal method or thermal decomposition technique. We mention that the method of obtaining

nanoparticles by thermal decomposition of an iron precursor in the presence of NaBH4 in a polyol was

found suitable for size control in both chemical approaches [1, 5].

Because certain technological procedures for the synthesis of these nanoparticles can affect

their quality in terms of dimensions, chemical composition and crystalline structure, the attention is

3

now directed towards improving their production quality. In this respect, Plackett-Burman design of

experiments is a filtration analysis used in the initial investigation of the main effects of the factors on a

characteristic of the final material [6].

It can be shown experimentally that the uncoated superparamagnetic nanoparticles are difficult

to synthesize, they cannot maintain stability in a colloidal suspension and, therefore, they are difficult

to use in the magnetic hyperthermia therapy [7].

By exposing the nanoparticles to the acidic environments of the living organisms, certain

processes of structural degradation caused by corrosion on the nanoparticle surfaces occur. This

biodegradation in acidic media leads over time to significant modification of the magnetic properties of

the nanoparticles [8]. These arguments are required to support the need to cover the nanoparticles. The

coating of nanoparticles is made for: reducing the nanoparticle surface sensitivity to air, humidity and

acidity, subsequent functionalization and absorption of proteins, creation of surface hydrophilic

molecules to prevent nanoparticle agglomeration, reduction of capillary obstruction risk, and improving

the blood circulation for nanoparticle transport in the targeted area, preservation of the physical-

chemical properties of the nanoparticles, prevention of nanoparticle opsonization by the

reticuloendothelial system that would cause their fast flowing from the bloodstream before reaching the

target and making a biocompatible and non-toxic shield around the nanoparticles, because their surface

is in direct contact with the blood and tissues. The thickness of the nanoparticle coating can

increasingly affect the magnetic properties, as well as the hyperthermia.

The materials used as a coating agent for the magnetic nanoparticles can be organic or

inorganic. The inorganic coating causes the surface of the nanoparticles to bind the biological ligands,

while maintaining the nanoparticle stability. The organic coating, in particular the polymer, has a

number of advantages, such as: better particle dispersion, good colloidal stability, biocompatibility,

good nanoparticle circulation in the blood, reduced toxicity and low risk of blood capillary obstruction.

4

In the recent years, a new class of nanofluids with biomedical applications, which are

ultrastable and often biocompatible, have been developed using a combination of electrostatic and

steric stabilisation methods [9]. Despite these stabilization methods, a number of recent papers have

experimentally shown a tendency to agglomeration, even in the absence of an external magnetic field

[10, 11]. This could be a problem when using ferrofluids for medical applications: the nanoparticle

agglomeration and sedimentation can create thrombi inside the blood vessels [12].

The control of agglomeration is becoming essential to improve the applicability of the magnetic

nanoparticles. In this regard, the optimized microemulsion method can be used to obtain

homogenous silica coatings on the Fe3−xO4 nanoparticles [13]. It show that the thickness of this coating

layer is controlled almost at will allowing much higher average separation among particles as compared

to the oleic acid coating present on pristine NPs [13].

Homogeneous, spherical polymer-coated magnetite nanoparticles have been successfully

prepared, showing a superparamagnetic response of the synthesized MNPs. The polymer coating has

been demonstrated to provide extra stability of the MNPs in aqueous media [14]. For better

biocompatibility or for specific hydrophilic properties, in most cases the nanoparticles are coated with

a polymer layer, such as the polyethylene glycol (PEG) [15].

The experimental data on the nanoparticle coating influence on their magnetic properties are

quite controversial. Recently, in situ reduction method applied for core-shell Fe3O4 with Ag and

polyacrylic acid indicated that the thin polymer coating layer had a great influence in enhancing the

hyperthermia efficiency [16]. Zavisova et al. [17] prepared and experimentally investigated the Fe3O4

nanoparticles with different protein or polymer outer shell. They found no correlation between the

coating layer thickness and the magnetic hyperthermia specific adsorption rate.

The issues mentioned in the previous paragraph show the need for research on the influence of

the coating on the magnetic behaviour of the nanoparticles, as other authors pointed out in their work

(e.g. [7]). To meet these needs, the current paper aims to use certain simulation models to study the

5

influence of nanoparticle coating on the tendency of nanoparticle agglomeration and on the Néel

magnetic relaxation time and, respectively, on the effective magnetic relaxation time.

Results and Discussion

Simulation methods used in the study

For numerical simulation, we start from a single-domain system, made of spherical iron oxide

nanoparticles, with a lognormal distribution of the grain sizes. Each nanoparticle is composed of a

magnetic core and a nonmagnetic surface layer of stabilizing surfactant. The system temperature is

considered constant.

For numerical simulation, two models developed by the authors and widely presented in the

specialized literature have been used [18- 20].

For self-organization of the colloidal magnetic nanoparticles simulation, we used the Langevin

dynamics stochastic method based on an effective Verlet-type algorithm [18].

The Néel magnetic relaxation time is assessed through the Coffey method in oblique magnetic

field, adapted to the local magnetic field on a nanoparticle [20- 21].

Method for self-organization of colloidal magnetic nanoparticles simulation

This method starts from numerical solutions of the Langevin equations for the translational and

rotational motions of an ith nanoparticle in the basic liquid [18].

The forces acting among the nanoparticles of the system, i.e. the Van der Waals forces, the

electrostatic repulsive forces, the magnetic dipolar forces, the steric repulsion forces have to be taken

into account for the coated nanoparticles and the random Brownian force [18, 22-25]. The stabilisation

of magnetic particles can be achieved by two repulsive forces: electrostatic and steric repulsion [18],

[22-23]. Besides the random Brownian torque, the conservative torque acting on the nanoparticle is a

magnetic torque. The random Brownian force and torque are usually modelled using the Gaussian

noise [20-21].

6

For solving numerically the equations of motion, we use the effective Verlet-type algorithm [18

- 19].

The Coffey method in oblique magnetic field, adapted to the local magnetic field on a

nanoparticle

In general, in the literature, the evaluation of the Néel relaxation time is made using the Néel-

Brown model [26], [27], valid if the nanoparticles do not interact magnetostatically one with the other

one. If the magnetic dipolar interaction among the nanoparticles is not neglected, this model leads to a

rather unrealistic approximation [20].

The dipolar magnetic field on the nanoparticle and the external magnetic field create a resultant

for internal magnetic field iH

on a nanoparticle, field that does not generally act along the direction of

easy magnetisation axis of the nanoparticle, being a so-called oblique magnetic field [21].

The nanoparticle Néel relaxation time in oblique magnetic fields is [20-21]:

21iE)2(2i

)2(1i

12iE)1(2i

)1(1i

12i

11iN0

iiN

eccecc

SS4

(1)

where Ei12 and Ei21 are the normalized energy barriers for magnetic moment re-orientations. The free

diffusion time of magnetisation N0i for low damping is [20-21]:

Tk2

Mv

B

siN0

i

(2)

where vi is the volume of the ith nanoparticle, Ms the spontaneous magnetisation, kB the Boltzmann

constant, T the temperature, the damping constant and the gyromagnetic ratio.

In Eq. (1):

iipiipi)p(

1i cosh2(cos2c , )cos(h(cos2c iipiip2

i)p(

2i , with p = 1, 2 (3)

Tk

vK=

B

ieffi

i (4)

7

with i the angle between iH

and the easy anisotropy axis of the ith nanoparticle. ip are the solutions

of the transcendental equation:

iiii sinh22sin (5)

In (5) i is the angle between magnetisation and easy anisotropy axis of the ith nanoparticle.

effi

isi

K

HMh

2

0 (6)

In Eq. (4) and (6) effiK effective magnetic anisotropy constant of the ith nanoparticle.

If hi < hic(Ψi) < 1 [20-21]:

...sin4cos192cos15618364

h

cossin2cos32

hsin2cos1113

16

hcossinhsin

2

h1cos

i2

ii

6i

ii2

i

5i

i2

i

4i

ii23

ii2

2i

2,1i

(7)

...]2sin2

h3sin3sin73cos3cos72sin

32

h

2sin2

hsincos2sin

2

hhcossinh21[E

i2

6i

iiiii

5i

i2

4i

iii

3i2

iiiii12i

(8)

...]2sin2

h3sin3sin73cos3cos72sin

32

h

2sin2

hsincos2sin

2

hhcossinh21[E

i2

6i

iiiii

5i

i2

4i

iii

3i2

iiiii21i

(9)

...sinh2sinh342sinh2

...]2cos5145sin2

h2cos211hsinh

3

10416[sinhS

i22

iiii2ii

ii

3i

i2

iiiii2,1i

(10)

Simulation conditions and results

For this study, we consider the case of a colloid with electrosteric stabilisation. The system

comprises 50 spherical magnetite nanoparticles with spontaneous magnetisation of Ms= 4.46105 A/m,

whose sizes have a lognormal distribution, which are dispersed in water whose dynamic viscosity is

8.910-4 Pas and the relative electrical permittivity is 78.5. The Hamaker constant for magnetite in

water is 3910-20 J [19]. The temperature is 298 K, the coating layer thickness ranges from 1 nm to 3

8

nm, the ion concentration in solution is 1026 ions/m3, their valence is 1, the surface density of the

polymers ranges from 1016 m-2 to 4.51017 m-2, and the surface charge is 1.610-15 C. The average

diameter of the nanoparticles is dm = 10 nm, with a standard deviation of 0.1dm. We consider that the

volume fraction of the nanoparticles is 0.05. We set the external magnetic field intensity to the value of

15kA/m along the Z-axis.

It is initially considered a random arrangement the nanoparticles in face-centred cubic grid.

With Langevin dynamics stochastic method, it is obtained an aggregate structure. For example, the

Figures 1-2 show the positions of nanoparticles inside the test cube a) at the initial moment and b) after

0.1ms (Figure 1 - the coating layer thickness is 1 nm and the surface density of the polymers = 1016

m-2, Figure 2 - the coating layer thickness is 1 nm and the surface density of the polymers = 4.51017

m-2).

0

2.27779e-08

4.55559e-08

6.83338e-08

9.11118e-08

0

2.27779e-08

4.55559e-08

6.83338e-08

9.11118e-08

0

2.27779e-08

4.55559e-08

6.83338e-08

9.11118e-08

X axis [m]

Positions of nanoparticles inside the test cube at simulation time 0.000e+00 [ s]

Y axis [m]

Z a

xis

[m

]

0

2.27779e-08

4.55559e-08

6.83338e-08

9.11118e-08

0

2.27779e-08

4.55559e-08

6.83338e-08

9.11118e-08

0

2.27779e-08

4.55559e-08

6.83338e-08

9.11118e-08

X axis [m]

Positions of nanoparticles inside the test cube at simulation time 1.000e+02 [ s]

Y axis [m]

Z a

xis

[m

]

a) b)

Fig. 1 - Positions of nanoparticles inside the test cube in a) initial moment and b) after 0.1ms (the

coating layer thickness is 1 nm, and the surface density of the polymers = 1016 m-2)

9

0

2.2965e-08

4.59299e-08

6.88949e-08

9.18599e-08

0

2.2965e-08

4.59299e-08

6.88949e-08

9.18599e-08

0

2.2965e-08

4.59299e-08

6.88949e-08

9.18599e-08

X axis [m]

Positions of nanoparticles inside the test cube at simulation time 0.000e+00 [ s]

Y axis [m]

Z a

xis

[m

]

0

2.2965e-08

4.59299e-08

6.88949e-08

9.18599e-08

0

2.2965e-08

4.59299e-08

6.88949e-08

9.18599e-08

0

2.2965e-08

4.59299e-08

6.88949e-08

9.18599e-08

X axis [m]

Positions of nanoparticles inside the test cube at simulation time 1.000e+02 [ s]

Y axis [m]

Z a

xis

[m

]

Fig. 2 - Positions of nanoparticles inside the test cube in a) initial moment and b) after 0.1ms (the

coating layer thickness is 1 nm, and the surface density of the polymers = 4.51017 m-2)

We can see in the figures 1 and 2, that the polymer concentration in the nanoparticle coating

influences the aggregate structure of the nanoparticles.

After obtaining the aggregate structure of the nanoparticles, by using the Coffey method in

oblique magnetic field, adapted to the local magnetic field on a nanoparticle, we can find, for each

nanoparticle, the Néel relaxation times iN . For each nanoparticle, the effective magnetic relaxation

time can be described as follows [27]:

iB

iN

ieff

111

(11)

where iB is the Brownian relaxation time. The Brownian process represents the nanoparticle rotation

in the fluid environment. For spherical particles, the Brownian relaxation time is usually described by:

Tk

v3

B

iHi

B

(12)

where vH is the hydrodynamic volume of the particle, and is the coefficient of dynamic viscosity.

10

Having for each nanoparticle the value of the effective magnetic relaxation time, we can

calculate the average effective magnetic relaxation time.

The relaxation mechanism which dominates the magnetic behaviour of the colloidal

suspensions is determined by the nanoparticle properties. For nanoparticles of 10 nm in diameter, it can

be shown that the Néel relaxation process is dominant (Figures 3, 4).

To study how the nanoparticle coating layer thickness and the polymer surface density

influence the magnetic behaviour of the nanoparticles, for different values of the polymer surface

density, we will vary the nanoparticle coating layer thickness between 1 nm and 3 nm. Then, for each

thickness of the coating layer, we will vary the polymer surface density between 1016 m-2 and 4.51017

m-2. The results are shown in Figures 3 and 4 a) and b). As can be seen in Figures 3 and 4, the average

Néel relaxation time and, respectively, the average effective magnetic relaxation time, are affected

either by the thickness of the nanoparticle coating layer or by the density of the polymer surface layer.

The Figures 3a) and b) show the average Néel relaxation time and, respectively, the average

effective magnetic relaxation time versus the nanoparticle coating layer thickness, at different values of

the polymer surface density.

Fig. 3 Average Néel relaxation time (a) and average effective magnetic relaxation time (b)

versus the nanoparticle coating layer thickness, at different values of the polymer surface density

11

In the Figures 3a) and b), we can see that, for low values of the polymers surface density in the

nanoparticle coating layer (1016 m-2 and 51016 m-2, respectively), the average Néel relaxation time (a)

and the average effective magnetic relaxation time (b) decreases with increasing layer thickness. At

values of about 1017 m-2 of the polymer surface density, the average Néel relaxation time (a) and the

average effective magnetic relaxation time (b) increases with decreasing coating layer thickness,

reaches a minimum value, followed by a slight increase. At high values of the polymer surface density,

the average Néel relaxation time (a) and the average effective magnetic relaxation time (b) increases

with increasing coating layer thickness, reaches a maximum value, followed by a slight decrease. The

obtained results can be explained by the competition between the attraction forces, especially the

magnetic dipolar interaction forces, directly proportional to the magnetic moments of the particles and

inversely proportional to the 5th power of the interparticle distances and the forces of rejection,

especially the steric forces directly proportional to the thickness of the surfactant coating layer. At low

values of the polymer surface density, the action of the rejection forces, in particular of the steric

forces, is weaker, and the forces of attraction predominate, especially the magnetic dipolar interaction

forces acting among the nanoparticles. As such, there is a tendency of agglomeration, which means

large local volumetric concentrations of large nanoparticles. At high values of the polymer surface

density, the action of the rejection forces, in particular the steric forces, is stronger, to the detriment of

the attraction forces, especially the forces of magnetic dipolar interaction acting among the

nanoparticles, which results in smaller local nanoparticle concentrations. In the extreme points

(minimum, maximum), an unstable balance is established between the forces of rejection and the forces

of attraction. Regarding the magnetic behaviour of the superparamagnetic nanoparticle system, the

literature shows a relaxation time decrease with increasing interparticle interaction in the case of

diluted systems [28-31]. On the other hand, it is claimed that the relaxation time increases when the

particle concentration increases [32-33].

12

Fig. 4 Average Néel relaxation time (a) and average effective magnetic relaxation time (b)

versus polymer surface density, at different values of the nanoparticle coating layer thickness

The Figures 4a) and b) show the Néel relaxation time and the effective magnetic relaxation

time versus the polymer surface density, at different thicknesses of the nanoparticle coating layer.

Regardless of the coating layer thickness, for low values of the polymer surface density, the average

Néel relaxation time and, respectively, the average effective magnetic relaxation time decreases with

increasing polymer surface density, reaches a minimum value, and then, at high values of the polymer

surface density, there is an increase in the average Néel relaxation time and, respectively, in the

average effective magnetic relaxation time. The minimum value of the average Néel relaxation time, as

well as the minimum value of the average effective magnetic relaxation time, is higher and higher as

the nanoparticle surfactant coating layer is thicker. For relatively small thicknesses of the nanoparticle

coating layer, the minimum value of the average Néel relaxation time, as well as the minimum value of

the average effective magnetic relaxation time, moves to small values of the polymer surface density,

as the thickness of the surfactant coating layer of the magnetic nanoparticles increases. For large

thicknesses, this displacement is almost unnoticeable. This complicated dependence can also be

explained by the competition between the rejection forces, in particular the steric rejection forces, and

13

the attraction forces, in particular the magnetic dipolar interaction forces acting among the

nanoparticles.

Conclusions

In this paper, two simulation models are used to understand how the surfactant coating layer

thickness and, respectively, the polymer surface layer density, are influencing the Néel relaxation time

and, respectively, the effective magnetic relaxation time, in a system of magnetic nanoparticles

suspended in a liquid matrix. So, for simulating the self-organization of the colloidal nanoparticles, we

used a Langevin dynamics stochastic method based on an effective Verlet-type algorithm, and for the

Néel relaxation time simulation we used the Coffey solution in oblique magnetic field adapted to the

local magnetic field on the nanoparticle, relation (1). The effective magnetic relaxation time was

calculated based on the relation (11).

Following the numerical simulations, it was revealed that the average Néel relaxation time and,

respectively, the average effective magnetic relaxation time, are affected either by the surfactant

coating layer thickness or by the polymer surface layer density.

More specifically, for small values of the polymer surface layer density in the nanoparticle

coating, the average Néel relaxation time and, respectively, the average effective magnetic relaxation

time decreases with increasing coating thickness. At intermediate values of the polymer surface layer

density, the average Néel relaxation time and, respectively, the average effective magnetic relaxation

time decreases with increasing the thickness of the coating, reaches a minimum value, after which a

slight increase occurs. At high values of the polymer surface layer density, the average Néel relaxation

time and, respectively, the average effective magnetic relaxation time increases with increasing the

thickness of the coating, reaches a maximum value, after which a slight decrease occurs.

It was also revealed that, regardless of the thickness of the coating, for small values of the

polymer surface layer density, the average Néel relaxation time and, respectively, the average effective

14

magnetic relaxation time decreases with the increase of the polymer surface layer density, reaches a

minimum value, after which, at high values of the polymer surface layer density, we notice an increase

of the average Néel relaxation time, as well of the average effective magnetic relaxation time. The

minimum value of the average Néel relaxation time and, respectively, the average effective magnetic

relaxation time, is higher and higher when the thickness of the nanoparticle surfactant coating is

increasing. For relatively small thicknesses of the nanoparticle coating layer, the minimum value of the

average Néel relaxation time and, respectively, of the average effective magnetic relaxation time,

moves to low values of the polymer surface density when the thickness of the surfactant coating of the

magnetic nanoparticles increases. For large thicknesses, this displacement is almost unnoticeable.

All these behaviours of the average Néel relaxation time and, respectively, of the average

effective magnetic relaxation time, can be explained by the competition between the rejection forces

and the attraction forces acting among the nanoparticles.

The results presented in this paper are of real help for various applications of the colloidal

magnetic nanoparticle systems, in particular biomedical applications [34-39], such as magnetic

hyperthermia with nanoparticles, which are of great interest either theoretically or applicatively, being

a possible alternative in the therapy of cancer in its various stages and forms.

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